Harmonic functions and volume growth of fingerless ends (Q1591404)

The author considers a complete, noncompact Riemannian manifold \(X\) of dimension greater than or equal to 2 without boundary. It is also supposed that \(X\) has finitely many ends, or equivalently, that the number of ends with respect to any compact set has a uniform upper bound. The author investigates the existence of the Green function for the \(p\)-Laplacian operator on \(M,\) expressed in terms of a suitable volume growth. In this note it is proved that the dimension of the linear space of polynomial growth harmonic functions is finite if a volume condition and an inequality for subharmonic functions hold in an adequate part of each end of the manifold \(X.\)